# std::ellint_1, std::ellint_1f, std::ellint_1l

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Defined in header `<cmath>` |  |
| (1) |
float ellint_1 ( float k, float phi ); double ellint_1 ( double k, double phi ); long double ellint_1 ( long double k, long double phi ); |  | (since C++17) (until C++23)
/* floating-point-type */ ellint_1( /* floating-point-type */ k, /* floating-point-type */ phi ); |  | (since C++23)
float ellint_1f( float k, float phi ); | (2) | (since C++17)
long double ellint_1l( long double k, long double phi ); | (3) | (since C++17)
Additional overloads |  |
Defined in header `<cmath>` |  |
template< class Arithmetic1, class Arithmetic2 > /* common-floating-point-type */ ellint_1( Arithmetic1 k, Arithmetic2 phi ); | (A) | (since C++17)

1-3) Computes the incomplete elliptic integral of the first kind of k and phi. The library provides overloads of `std::ellint_1` for all cv-unqualified floating-point types as the type of the parameters k and phi.(since C++23)
A) Additional overloads are provided for all other combinations of arithmetic types.

### Parameters

k | - | elliptic modulus or eccentricity (a floating-point or integer value)
phi | - | Jacobi amplitude (a floating-point or integer value, measured in radians)

### Return value

If no errors occur, value of the incomplete elliptic integral of the first kind of k and phi, that is ∫phi
0

dθ
√1-k2 sin2 θ

, is returned.

### Error handling

Errors may be reported as specified in math_errhandling:

- If the argument is NaN, NaN is returned and domain error is not reported.
- If |k|>1, a domain error may occur.

### Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if `__STDCPP_MATH_SPEC_FUNCS__` is defined by the implementation to a value at least 201003L and if the user defines `__STDCPP_WANT_MATH_SPEC_FUNCS__` before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header `tr1/cmath` and namespace `std::tr1`.

An implementation of this function is also available in boost.math.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their first argument num1 and second argument num2:

If num1 or num2 has type long double, then std::ellint_1(num1, num2) has the same effect as std::ellint_1(static_cast<long double>(num1), static_cast<long double>(num2)). Otherwise, if num1 and/or num2 has type double or an integer type, then std::ellint_1(num1, num2) has the same effect as std::ellint_1(static_cast<double>(num1), static_cast<double>(num2)). Otherwise, if num1 or num2 has type float, then std::ellint_1(num1, num2) has the same effect as std::ellint_1(static_cast<float>(num1), static_cast<float>(num2)). | (until C++23)
If num1 and num2 have arithmetic types, then std::ellint_1(num1, num2) has the same effect as std::ellint_1(static_cast</* common-floating-point-type */>(num1), static_cast</* common-floating-point-type */>(num2)), where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2, arguments of integer type are considered to have the same floating-point conversion rank as double. If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided. | (since C++23)

### Example

Run this code

    #include <cmath>
    #include <iostream>
    #include <numbers>

    int main()
    {
        const double hpi = std::numbers::pi / 2.0;

        std::cout << "F(0,π/2)  = " << std::ellint_1(0, hpi) << '\n'
                  << "F(0,-π/2) = " << std::ellint_1(0, -hpi) << '\n'
                  << "π/2       = " << hpi << '\n'
                  << "F(0.7,0)  = " << std::ellint_1(0.7, 0) << '\n';
    }

Output:

    F(0,π/2)  = 1.5708
    F(0,-π/2) = -1.5708
    π/2       = 1.5708
    F(0.7,0)  = 0

### See also

comp_ellint_1comp_ellint_1fcomp_ellint_1l (C++17)(C++17)(C++17) | (complete) elliptic integral of the first kind (function)

### External links

Weisstein, Eric W. "Elliptic Integral of the First Kind." From MathWorld — A Wolfram Web Resource.

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